Lp space - Citizendia

The correct title of this article is L^p space. It features superscript or subscript characters that are substituted or omitted because of technical limitations. This article is about the terms 'subscript' and 'superscript' as used in typography This article is about the terms 'subscript' and 'superscript' as used in typography

In mathematics, the L^p and $\ell^p$ spaces are spaces of p-power integrable functions, and corresponding sequence spaces. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, an integrable function is a function whose Integral exists In Functional analysis and related areas of Mathematics, a sequence space is a Vector space whose elements are infinite Sequences of Complex They are sometimes called Lebesgue spaces, named after Henri Lebesgue (Dunford & Schwartz 1958, III. Henri Léon Lebesgue leɔ̃ ləˈbɛg ( June 28, 1875, Beauvais &ndash July 26, 1941, Paris) was a French 3). They form an important class of examples of Banach spaces in functional analysis, and of topological vector spaces. In Mathematics, Banach spaces (ˈbanax named after Polish Mathematician Stefan Banach) are one of the central objects of study in Functional analysis For functional analysis as used in psychology see the Functional analysis (psychology article In Mathematics, a topological vector space is one of the basic structures investigated in Functional analysis. Lebesgue spaces have applications in physics, statistics, finance, engineering, and other disciplines.

1 Motivation
2 ℓ^p spaces
3 Properties of ℓ^p spaces
4 L^p spaces
- 4.1 Special cases
5 Properties of L^p spaces
- 5.1 Dual spaces
- 5.2 Embeddings
6 Applications
7 Weighted L^p spaces
8 See also
9 References
10 External links

Motivation

Consider the real vector space Rⁿ. In Mathematics, the real numbers may be described informally in several different ways In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added The sum of vectors in Rⁿ is given by

$\ (x_1, x_2, \dots, x_n) + (y_1, y_2, \dots, y_n) = (x_1+y_1, x_2+y_2, \dots, x_n+y_n),$

and the scalar action is given by

$\ \lambda(x_1, x_2, \dots, x_n)=(\lambda x_1, \lambda x_2, \dots, \lambda x_n).$

The length of a vector $x=(x_1, x_2, \dots, x_n)$ is usually given by

$\ \|x\|_2=\left(x_1^2+x_2^2+\dots+x_n^2\right)^{1/2}$

but this is by no means the only way of defining length. If p is a real number, p ≥ 1, define

$\ \|x\|_p=\left(|x_1|^p+|x_2|^p+\dots+|x_n|^p\right)^{1/p}$

for any vector $x=(x_1, x_2, \dots, x_n)$ . In Mathematics, the real numbers may be described informally in several different ways It turns out that this definition indeed satisfies the properties of a "length function" (or norm), which are that only the zero vector has zero length, the length of the vector changes (modulus-)linearly when we multiply it by a scalar, and the length of the sum of two vectors is no larger than the sum of lengths of the vectors (triangle inequality). In Linear algebra, Functional analysis and related areas of Mathematics, a norm is a function that assigns a strictly positive length In Mathematics, the triangle inequality states that for any Triangle, the length of a given side must be less than or equal to the sum of the other two sides but greater For any p ≥ 1, Rⁿ together with the p-norm just defined becomes a Banach space. In Mathematics, Banach spaces (ˈbanax named after Polish Mathematician Stefan Banach) are one of the central objects of study in Functional analysis

ℓ^p spaces

The above p-norm can be extended to vectors having an infinite number of components, yielding the space $\ell^p$ . For $\ x=(x_1, x_2, \dots, x_n, x_{n+1},\dots)$ an infinite sequence of real (or complex) numbers, define the vector sum to be

$\ (x_1, x_2, \dots, x_n, x_{n+1},\dots)+(y_1, y_2, \dots, y_n, y_{n+1},\dots)=(x_1+y_1, x_2+y_2, \dots, x_n+y_n, x_{n+1}+y_{n+1},\dots),$

while the scalar action is given by

$\ \lambda(x_1, x_2, \dots, x_n, x_{n+1},\dots) = (\lambda x_1, \lambda x_2, \dots, \lambda x_n, \lambda x_{n+1},\dots).$

Define the p-norm

$\ \|x\|_p=\left(|x_1|^p+|x_2|^p+\dots+|x_n|^p+|x_{n+1}|^p+\dots\right)^{1/p}.$

Here, a complication arises, namely that the series on the right is not always convergent, so for example, the sequence made up of only ones, $(1, 1, 1, \dots),$ will have an infinite p-norm (length) for every finite p ≥ 1. In Mathematics, a sequence is an ordered list of objects (or events Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In Mathematics, a series is often represented as the sum of a Sequence of terms That is a series is represented as a list of numbers with The space $\ell^p$ is then defined as the set of all infinite sequences of real (or complex) numbers such that the p-norm is finite.

One can check that as p increases, the set $\ell^p$ grows larger. For example, the sequence

$\ \left(1, \frac{1}{2}, \dots, \frac{1}{n}, \frac{1}{n+1},\dots\right)$

is not in $\ell^1$ , but it is in $\ell^p$ for p>1, as the series

$\ 1^p+\frac{1}{2^p} + \dots + \frac{1}{n^p} + \frac{1}{(n+1)^p}+\dots$

diverges for p=1 (the harmonic series), but is convergent for p>1. See Harmonic series (music for the (related musical concept In Mathematics, the harmonic series is the Infinite series

One also defines the ∞-norm as

$\ \|x\|_\infty=\sup(|x_1|, |x_2|, \dots, |x_n|,|x_{n+1}|, \dots)$

and the corresponding space $\ell^\infty$ of all bounded sequences. It turns out that

$\ \|x\|_\infty=\lim_{p\to\infty}\|x\|_p$

if the right-hand side is finite, or the left-hand side is infinite. Thus, we will consider $\ell^p$ spaces for 1≤p≤∞.

The p-norm thus defined on $\ell^p$ is indeed a norm, and $\ell^p$ together with this norm is a Banach space. The fully general L^p space is obtained — as seen below — by considering vectors, not only with finitely or countably-infinitely many components, but with arbitrarily many components; in other words, functions. The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function An integral instead of a sum is used to define the p-norm. The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space

Properties of ℓ^p spaces

The space $\ell^2$ is the only $\ell^p$ space that is a Hilbert space, since any norm that is induced by an inner product should satisfy the parallelogram identity $\|x+y\|_p^2 + \|x-y\|_p^2= 2\|x\|_p^2 + 2\|y\|_p^2$ . This article assumes some familiarity with Analytic geometry and the concept of a limit. Substituting two distinct unit vectors in for x and y directly shows that the identity is not true unless p = 2.

The $\ell^p$ , 1 < p < ∞ spaces are reflexive: $(\ell^p)^*=\ell^q$ , where (1/p) + (1/q) = 1. In Functional analysis, a Banach space is called reflexive if it satisfies a certain abstract property involving Dual spaces Reflexive spaces turn out to

The dual of c₀ is $\ell^1$ ; the dual of $\ell^1$ is $\ell^\infty$ . For the case of natural numbers index set, the $\ell^p$ and c₀ are separable, with the sole exception of $\ell^{\,\infty}$ . In Mathematics a Topological space is called separable if it contains a countable dense subset that is there exists a sequence \{ x_n Here, c₀ is defined as the space of all sequences converging to zero, with norm identical to ||x||_∞. The dual of $\ell^{\,\infty}$ is the ba space. In Mathematics, the ba space ba(\Sigma of an Algebra of sets \Sigma is the Banach space consisting of all bounded

The $\ell^p$ spaces can be embedded into many Banach spaces. In Mathematics, an embedding (or imbedding) is one instance of some Mathematical structure contained within another instance such as a group In Mathematics, Banach spaces (ˈbanax named after Polish Mathematician Stefan Banach) are one of the central objects of study in Functional analysis The question of whether all Banach spaces have such an embedding was answered negatively by B. S. Tsirelson's construction of Tsirelson space in 1974. Boris Semyonovich Tsirelson (בוריס סמיונוביץ' צירלסון Борис Семенович Цирельсон is a Soviet - Israeli Mathematician In Mathematics, Tsirelson space T is an example of a reflexive Banach space in which neither an ''l p'' space nor a ''c''0 Year 1974 ( MCMLXXIV) was a Common year starting on Tuesday (link will display full calendar of the 1974 Gregorian calendar.

Except for the trivial finite case, an unusual feature of $\ell^p$ is that it is not polynomially reflexive. In Mathematics, a polynomially reflexive space is a Banach space X, on which all polynomials are reflexive.

L^p spaces

Let 1 ≤ p < ∞ and (S, μ) be a measure space. In Mathematics the concept of a measure generalizes notions such as "length" "area" and "volume" (but not all of its applications have to do with Consider the set of all measurable functions from S to C (or R) whose absolute value raised to the p-th power has a finite Lebesgue integral, or equivalently, that

$\|f\|_p := \left({\int |f|^p\;\mathrm{d}\mu}\right)^{1/p}<\infty.$

The set of such functions form a vector space, with the following natural operations:

$(f+g)(x)=f(x)+g(x) \,$

and, for a scalar λ,

$(\lambda f)(x) = \lambda f(x). \,$

That the sum of two p^th power integrable functions is again p^th power integrable follows from the inequality |f + g|^p ≤ 2^p (|f|^p + |g|^p). In Mathematics, measurable functions are Well-behaved functions between measurable spaces. In Mathematics, the absolute value (or modulus) of a Real number is its numerical value without regard to its sign. In Mathematics, the Integral of a non-negative function can be regarded in the simplest case as the Area between the graph of In fact, more is true. Minkowski's inequality says the triangle inequality holds for $\| \cdot \|_p.$

Thus the set of p^th power integrable functions, together with the function $\| \cdot \|_p$ , is a seminormed vector space, which we denote by $\mathcal{L}^p(S, \mu).$

This can be made into a normed vector space in a standard way; one simply takes the quotient space with respect to the kernel of ||·||_p. In Mathematical analysis, the Minkowski inequality establishes that the L''p'' spaces are Normed vector spaces Let S be a Measure In Linear algebra, Functional analysis and related areas of Mathematics, a norm is a function that assigns a strictly positive length Since ||f||_p = 0 if and only if f = 0 almost everywhere, in the quotient space two functions f and g are identified if f = g almost everywhere. In Measure theory (a branch of Mathematical analysis) one says that a property holds almost everywhere if the set of elements for which the property does The resulting normed vector space is, by definition,

$L^p(S, \mu) := \mathcal{L}^p(S, \mu) / \mathrm{ker}(\|\cdot\|_p) .$

For p = ∞, the space L^∞(S, μ) is defined as follows. We start with the set of all measurable functions from S to C (or R) which are essentially bounded, i. e. bounded up to a set of measure zero. Again two such functions are identified if they are equal almost everywhere. Denote this set by L^∞(S, μ). For f in L^∞(S, μ), its essential supremum serves as an appropriate norm:

$\|f\|_\infty := \inf \{ C\ge 0 : |f(x)| \le C \mbox{ for almost every } x\}.$

As before, we have

$\|f\|_\infty=\lim_{p\to\infty}\|f\|_p$

if f ∈ L^∞(S,μ) ∩ L^q(S,μ) for some q < ∞.

For 1 ≤ p ≤ ∞, L^p(S, μ) is a Banach space. Completeness can be checked using the convergence theorems for Lebesgue integrals.

When the underlying measure space S is understood, L^p(S,μ) is often abbreviated L^p(μ), or just L^p. The above definitions generalize to Bochner spaces. In Mathematics, Bochner spaces are a generalization of the concept of ''Lp'' spaces to more general domains and ranges than the initial definition specifically

Special cases

When p = 2; like the $\ell^2$ space, the space L² is the only Hilbert space of this class. This article assumes some familiarity with Analytic geometry and the concept of a limit. The additional inner product structure allows for a richer theory, with applications to, for instance, Fourier series and quantum mechanics. In Mathematics, a Fourier series decomposes a periodic function into a sum of simple oscillating functions Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons

If we use complex-valued functions, the space L^∞ is a commutative C*-algebra with pointwise multiplication and conjugation. In Mathematics, commutativity is the ability to change the order of something without changing the end result C*-algebras (pronounced "C-star" are an important area of research in Functional analysis, a branch of Mathematics. For many measure spaces, including all sigma-finite ones, it is in fact a commutative von Neumann algebra. In Mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the An element of L^∞ defines a bounded operator on any L^p space by multiplication. In Functional analysis (a branch of Mathematics) a bounded linear operator is a Linear transformation L between Normed vector spaces In Operator theory, a multiplication operator is a Linear operator T defined on some vector space of functions and whose value at a function

The $\ell^p$ spaces (1 ≤ p ≤ ∞) are a special case of L^p spaces, when the set S is the positive integers, and the measure used in the integration in the definition is a counting measure. The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French In Mathematics, the counting measure is an intuitive way to put a measure on any set: the "size" of a Subset is taken to be the number More generally, if one considers any set S with the counting measure, the resulting L^p space is denoted $\ell^p(S)$ . For example, the space $\ell^p(\mathbb Z)$ is the space of all sequences indexed by the integers, and when defining the p-norm on such a space, one sums over all the integers. The space $\ell^p(n)$ , where n is the set with n elements, is Rⁿ with its p-norm as defined above.

Properties of L^p spaces

Dual spaces

The dual space (the space of all continuous linear functionals) of $L p (μ)$ for $1 < p < \infty$ has a natural isomorphism with $L q (μ)$ , where q is such that 1/p + 1/q = 1, which associates $g\in L^q$ with the functional $\kappa_g\in L^p(\mu)^*$ defined by

$\kappa_g(f) = \int f g \;\mbox{d}\mu.$

The mapping $\kappa : g\mapsto\kappa_g$ is a linear mapping from $L q (μ)$ into $L p (μ) *$ , which is an isometry onto its image by Hölder's inequality. In Mathematics, any Vector space V has a corresponding dual vector space (or just dual space for short consisting of all Linear functionals For the Mechanical engineering and Architecture usage see Isometric projection. In Mathematical analysis Hölder's inequality, named after Otto Hölder, is a fundamental Inequality between integrals and an indispensable tool It is also possible to show that any G $\in L^p(\mu)^*$ can be expressed this way: i. e. , that κ is a continuous linear bijection of Banach spaces. In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property By the open mapping theorem, it follows that κ is an isomorphism of Banach spaces. In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective

Since the relationship 1/p + 1/q = 1 is symmetric, $L p (μ)$ is reflexive for these values of p: the natural monomorphism from $L p (μ)$ to $L p (μ) * *$ obtained by composing κ with the adjoint of its inverse

$L^p(\mu) \overset{\kappa}{\to} L^q(\mu)^* \overset{\,\,(\kappa^{-1})^*}{\longrightarrow} L^p(\mu)^{**}$

is onto, that is, it is an isomorphism of Banach spaces. In Functional analysis, a Banach space is called reflexive if it satisfies a certain abstract property involving Dual spaces Reflexive spaces turn out to In the context of Abstract algebra or Universal algebra, a monomorphism is simply an Injective Homomorphism. In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective

If the measure μ on S is sigma-finite, then the dual of L¹(μ) is isomorphic to L^∞(μ). However, except in rather trivial cases, the dual of L^∞ is much bigger than L¹. Elements of (L^∞)^* can be identified with bounded signed finitely additive measures on S in a construction similar to the ba space. In Mathematics, the ba space ba(\Sigma of an Algebra of sets \Sigma is the Banach space consisting of all bounded

If 0 < p < 1, then L^p can be defined as above, but || · ||_p does not satisfy the triangle inequality in this case, and hence it defines only a quasi-norm. In Linear algebra, Functional analysis and related areas of Mathematics, a quasinorm is similar to a norm in that it satisfies the norm axioms However, we can still define a metric by setting d(f, g) = (||f − g||_p)^p. In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined The resulting metric space is complete, and L^p for 0 < p < 1 is the prototypical example of an F-space that is not locally convex. In Mathematical analysis, a Metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has In Functional analysis, an F-space is a Vector space V over the real or complex numbers together with a metric d In Functional analysis and related areas of Mathematics, locally convex topological vector spaces or locally convex spaces are examples of Topological

Embeddings

Colloquially, if 1 ≤ p < q ≤ ∞, L^p(S) contains functions that are more locally singular while elements of L^q(S) can be more spread out. Consider the Lebesgue measure on the half line (0, ∞). A continuous function in L¹ might blow up near 0 but must decay sufficiently fast toward infinity. On the other hand, continuous functions in L^∞ need not decay at all but no blow-up is allowed. The precise technical result is following:

L^p(S) is not contained in L^q(S) iff S contains sets of arbitrarily small measure, and

L^q(S) is not contained in L^p(S) iff S contains sets of arbitrarily large measure. In particular, if the domain S has finite measure, the bound (a consequence of Hölder's inequality)

$\ \|f\|_p \le \mu(S)^{(1/p)-(1/q)} \|f\|_q$

means the space L^q is continuously embedded in L^p. In Mathematical analysis Hölder's inequality, named after Otto Hölder, is a fundamental Inequality between integrals and an indispensable tool

Applications

$L p$ spaces are widely used in mathematics and applications.

Fourier series transform between $L p$ and $\ell^q$ . In Mathematics, a Fourier series decomposes a periodic function into a sum of simple oscillating functions

The Hilbert space $L 2$ is central to quantum mechanics. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons

In statistics, measures of central tendency and statistical dispersion, such as the mean, median, and standard deviation, are defined in terms of $L p$ metrics, and measures of central tendency can be characterized as solutions to variational problems. Statistics is a mathematical science pertaining to the collection analysis interpretation or explanation and presentation of Data. In Mathematics, an average, or central tendency of a Data set refers to a measure of the "middle" or " expected " value of In Statistics, (statistical dispersion (also called statistical variability or variation) is variability or spread in a Variable or a Probability In Statistics, mean has two related meanings the Arithmetic mean (and is distinguished from the Geometric mean or Harmonic mean In Probability theory and Statistics, a median is described as the number separating the higher half of a sample a population or a Probability distribution In Probability and Statistics, the standard deviation is a measure of the dispersion of a collection of values In Mathematics, an average, or central tendency of a Data set refers to a measure of the "middle" or " expected " value of

Weighted L^p spaces

As before, consider a measure space $(S, \mathcal{F}, \mu)$ . In Mathematics the concept of a measure generalizes notions such as "length" "area" and "volume" (but not all of its applications have to do with Let $w : S \to [0, + \infty)$ be a measurable function. The $w$ -weighted L^p space is defined as $L^{p} (S, w \, \mathrm{d} \mu)$ , where $w \, \mathrm{d} \mu$ means the measure $ν$ defined by

$\ \nu (A) := \int_{A} w(x) \, \mathrm{d} \mu (x),$

or, in terms of the Radon-Nikodym derivative,

$\ w = \frac{\mathrm{d} \nu}{\mathrm{d} \mu}.$

The norm for $L^{p} (S, w \, \mathrm{d} \mu)$ is explicitly

$\ \| u \|_{L^{p} (S, w \, \mathrm{d} \mu)} := \left( \int_{S} w(x) | u(x) |^{p} \, \mathrm{d} \mu (x) \right)^{1/p}.$

References

Adams, Robert A. In Mathematics, the Radon–Nikodym theorem is a result in Functional analysis that states that given a measurable space ( X,&Sigma if a In Linear algebra, Functional analysis and related areas of Mathematics, a norm is a function that assigns a strictly positive length In Complex analysis, the Hardy spaces (or Hardy classes) H p are certain spaces of holomorphic functions on the unit disk A generalized mean, also known as power mean or Hölder mean, is an abstraction of the Pythagorean means including arithmetic, geometric In Mathematics, a real-valued function f on R n satisfies a Hölder condition, or is Hölder continuous, when there are In Mathematics, the root mean square (abbreviated RMS or rms) also known as the quadratic mean, is a statistical measure of the In Mathematics, a locally integrable function is a function which is integrable on any Compact set of its domain of definition. (1975), Sobolev Spaces, New York: Academic Press, ISBN 0-12-044150-0
Dunford, Nelson & Schwartz, Jacob T. (1958), Linear operators, volume I, Wiley-Interscience .
Hewitt, Edwin & Stromberg, Karl (1965), Real and abstract analysis, Springer-Verlag .

External links

Proof that L^p spaces are complete on PlanetMath

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Contents

Motivation

ℓp spaces

Properties of ℓp spaces

Lp spaces